Error Contributions

The error contributions to an impedance measurement can expressed in terms of the difference between the observed value Zob co) and a model value Z nodi ) 

A distinction is drawn in equation (21.1) between stochastic errors that are randomly distributed about a mean value of zero, errors caused by the lack of fit of a model, and experimental bias errors that are propagated through the model. The problem of interpretation of impedance data is therefore defined to consist of two parts one of identification of experimental errors, which includes assessment of consistency with the Kramers-Kronig relations (see Chapter 22), and one of fitting (see Chapter 19), which entails model identification, selection of weighting strategies, and examination of residual errors. The error analysis provides information that can be incorporated into regression of process models. The experimental bias errors, as referred to here, may be caused by nonstationary processes or by instrumental artifacts. 

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Calculate human-errors contribution to probability of system failure... 

Human error contributed to about 50% of the accident sequences m the RSS but none of the human error data came from the nuclear power industry. Furthermore, very high failure rates 0.5 to 0.1/action) were predicted but are not supported by the plant... 

The primary goal of this series of chapters is to describe the statistical tests required to determine the magnitude of the random (i.e., precision and accuracy) and systematic (i.e., bias) error contributions due to choosing Analytical METHODS A or B, and/or the location/operator where each standard method is performed. The statistical analysis for this series of articles consists of five main parts as ... 

Note that for the calculations of precision and standard deviation (equations 38-1 through 38-4), the numerator expression is given as 2(n — 1). This expression is used due to the 2 times error contribution from independent errors found in each independent set (i.e., X and Y) of results. 

The systematic error contribution is larger for METHOD B than METHOD A. 

Ensure that the responses from samples are close to the mean response, y, of the calibration set. This will decrease the error contribution from the least-squares estimate of the regression line. 

The above measurement results also included the error contribution of the temperature cross-sensitivity of the device. From Fig. 7.11, the temperature dependence of the device was 0.074 nm °C Based on (7.6), the temperature crosssensitivity of the device was less than 3.2 x 10 6 RIU °C. Therefore, the total temperature cross-sensitivity-induced measurement error was about 2.8 x 10 4 RIU in Fig. 7.13 over the temperature variation of 87°C. The temperature dependence of the device was small and contributed only about 2.3% to the total refractive index variation over the entire temperature range. 

A modified Youden two sample quality control scheme is used to provide continuous analytical performance surveillance. The basic technique described by other workers has been extended to fully exploit the graphical identification of control plot patterns. Seven fundamental plot patterns have been identified. Simulated data were generated to illustrate the basic patterns in the systematic error that have been observed in actual laboratory situations. Once identified, patterns in the quality control plots can be used to assist in the diagnosis of a problem. Patterns of behavior in the systematic error contribution are more frequent and easy to diagnose. However, pattern complications in both error domains are observed. This paper will describe how patterns in the quality control plots assist interpretation of quality control data. 

Assume that a single sample is split into two portions labeled A and B. A quantitative determination of some sample constituent should yield the "true" value X plus any systematic and random error contributions. 

Since the random error contributions, R and R , have identical distributions symmetric about zero, and with expectation of zero an average value of T, based on a large number of observations will have a very small component from averaging of R and R. ... 

The primary purpose of any quality control scheme is to identify ("flag") significant performance changes. The two-sample quality control scheme described above effectively identifies performance changes and permits separation of random and systematic error contributions. It also permits rapid evaluation of a specific analytical result relative to previous data. Graphical representation of these data provide effective anomaly detection. The quality control scheme presented here uses two slightly different plot formats to depict performance behavior. 

Figure 2. Plots showing location of measured values with various systematic and random error contributions.

Figure 4) No systematic error is present. The only error contribution is the result of random deviations in the results obtained for the two quality control samples, A and B. 

An absence of large random error contribution corresponding to anomalous points in the T plot shows that they are in the systematic error domain. (This is more readily seen in the Q plot.)... 

Figure 6) The systematic error contribution obeys a step function. The absence of any systematic bias during the early time period is followed by a sudden appearance of a large constant systematic error (either positive or negative). 

Figure 7) The systematic error contribution increases or decreases monotonically with increasing time. 

Figure 8) The systematic error contribution increases or decreases rapidly with time, but finally levels off to a constant value. This behavior is similar to, but occurs less precipitously than, the step function exhibited by the SHIFT. 

Figure 9) The magnitude of the systematic error contribution changes continuously with time, but it follows a definite cyclic pattern that repeats itself periodically. (This case is simulated here as a sine wave.)... 

While only -10% of microarray datasets address the problem of batch effects (48), the degree of error contributed by batch effects may be significant. Batch effects may include experimental variations introduced due to multiple types of technical bias (e.g., time, laboratory, reagents, handling). Analysis of multiple methods to address batch effects has been addressed for precision, accuracy, and overall performance (48). Once probe set raw intensities have been processed via normalization and possible additional corrective measures, values can be used for downstream analyses in identifying differentially expressed genes and corresponding functional associations. 

Different sources of systematic errors contribute to the overall bias (Figure 8). Thompson and Wood [8] describe persistant bias as the bias affecting all data of the analytical system over longer periods of time and being relatively small but continuously present. Different components contribute to the persistant bias, such as laboratory bias, method bias, and the matrix variation effect. Next to persistant bias, the larger run effect is the bias of the analytical system during a particular run... 

As dimensions get smaller it is increasingly important to improve overlay and this places stringent requirements on the distortion of the lenses. The distortion of the best lenses remains at about 0.1 i m and this error contributes significantly to final overlay when different cameras are used for different layers. 

Since the actual data contains noise and computational roundoff errors, additional nonzero eigenvalues (noise eigenvalues) will be generated by the computation. The theory shows that the eigenvalues can be grouped into two sets a set which contains the factors or components together with an error contribution and a secondary set composed entirely of error. 

An estimate of the probable errors in the correction factors and cutoff values follows. From Equation 3 one sees that the fractional errors in both are of the same order of magnitude as the fractional error in the velocity, Av/v, averaged over the region of motion. There are three main contributions to this error. One comes from the approximation to the Davies equations (8 and 10). The average fractional error is of the order of Av/v —5%, the minus sign occurring since Equations 9 and 10 underestimate the true values of Re and v. The other error contributions come from the approximations for air density and viscosity. One sees from Equations 7-9 that the first-order term in v is independent of p and has a 1/rj dependence. The second-order term is directly proportional to P. Since this term contributes a maximum of 30% to the velocity and the maximum error in p is 8%, this contribution to Av/v should be... 

The focus of the previous section was to estimate the expected error from assuming the zonal invariance of mean values of moist static energy. This expected error contributes to the total expected error of a paleoaltitude estimate. Before proceeding to the next section (inferring paleoclimate from plant fossils), we examine the zonal invariance assumption as applied in the mean annual temperature approach to paleoaltimetry. Based on the initial method of Axelrod (1966), paleoaltitudes can be estimated by comparing mean annual temperature differences using the formula... 

What is the error contributed to the calculation when the counting efficiency obtained for water is applied to the counting efficiency for KC1 measurement ... 

Since j is functionally dependent on x, the uncertainty in / must contain a contribution from any uncertainty in x. If in the above-described procedure, the variable x is set separately and independently to its assigned value Xi for each of the A measurements / , then presumably the contribution of the uncertainty in Xi will automatically be reflected in the uncertainty of / and Eq. (23) will apply directly. However, if x is set only once to a for the Ameasurements /> so that the error contributed by Xi is the same for all Ameasure-ments, then the uncertainty in both must be reflected in the weight... 

Some of the concepts used in defining confidence limits are extended to the estimation of uncertainty. The uncertainty of an analytical result is a range within which the true value of the analyte concentration is expected to lie, with a given degree of confidence, often 95%. This definition shows that an uncertainty estimate should include the contributions from all the identifiable sources in the measurement process, i.e. including systematic errors as well as the random errors that are described by confidence limits. In principle, uncertainty estimates can be obtained by a painstaking evaluation of each of the steps in an analysis and a summation, in accord with the principle of the additivity of variances (see above) of all the estimated error contributions any systematic errors identified... 

Ultrasonic energy is frequently used to accelerate the dissolution of solid samples under soft conditions of temperature, pressure and chemical reagents. Similar to direct dissolution by agitation, US-assisted soft digestion is not used to the same extent as other operations of the analytical process such as leaching, derivatization or detection. The simplicity of this operation with some types of samples and the operator s lack of awareness of its error contribution are responsible for the absence of optimization studies for this process. Inappropriately conducted soft digestion can result in major errors and affect the quality of the results. 

The first steps in any sampling investigation are audit and assessment find out what is going on and whether the current sampling variation is acceptable. If not, then some way must be found to reduce it. This would be easier if the total variation could be broken down and the component parts addressed separately. Pierre Gy s theory does this. Gy (1992) decomposes the total variation into seven major components (sources). He calls them errors because sampling is an error-generating process, and these errors contribute to the nonrepresentativeness of the sample. The seven errors are as follows ... 

Delimitation and extraction errors contribute to both bias and variation, and bias is very difficult to detect unless a special effort is made. As a result, the magnitude of these errors is often unknown and frequently underestimated. It would be extremely unfortunate to learn of a bias via a lawsuit. To avoid bias, a proper tool must be chosen and used correctly. Even if the right equipment is available, additional error is added to the total sampling error if the equipment is not used correctly. 

Suppose that v 6) is block-diagonal and that E is unknown (though its relevant elements will form a block-diagonal array, as in Table 7.2b). Suppose that Y yields sample estimates v b with Vbe degrees of freedom for the experimental error contributions to one or more of the residual block matrices Vb 0j)- Then the posterior probabilities based on the combined data take the form... 

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